I just got back from a 5-day trip to Australia, and I envy you. In each of the cities I visited (Brisbane, Sydney and Melbourne), there are plenty of bike lanes. I can only wish my native New York is that bike-friendly.
That said, $250AUS isn't going to get you very far (I noticed that things are very expensive there). You can forget about an e-bike-- Even used e-bikes (that are in decent working shape) are going to cost A lot more than that. And you DO NOT want a heavy cheap department-store mountain bike because those are the WORST for climbing hills.
Get a road bike (which has no suspension) for tackling hills. Cheap heavy mountain bikes with suspensions will only waste your pedaling power through the suspension bouncing and make you tired faster.
Next time I visit Australia I'm going to ship my bike along. I want to climb that road that goes up to the Dandenong Ranges National Park outside Melbourne!
I'm not sure I agree about getting a road bike for the conditions you mentioned. Most of the roadies you'll find in your price range are likely to have what's become the standard gearing, something like 53-39 chainrings and a large cog (gear in the rear) of 26 or so. For a fat, lazy rider (I speak from experience here), that's too high for hills.
Weight is important, of course, but the difference between a heavy bike and a light one is only eight or 10 pounds. You're probably carrying more than that around your waist. A reasonably light mountain bike will give you a triple chainring and probably a larger cog in back, which will help more on climbs than the extra weight will hurt. Of course a road bike with triple rings would do the same, but they're hard to find.
You can also change the knobby tires on a mountain bike to skinnier, higher-pressure rubber, which will help a lot.